3 POSITIVE MEASURES: THE ABSOLUTE VALUE OF A MEASURE 105 The notion of a positive measure enables us to define an order relation on the vector space MR(X) of real measures on X. We write jj,^ v if the measure v — ^ is positive. Since the relations /x^O and n^Q imply that /*(/) = X/+) - X/~) = ° for all/e ^n(X), and therefore that ^ = 0, it follows that li <£ v is indeed an order relation on MR(X) (not, in general, a total ordering). It is clear that \JL <£ v implies that 1 + v ^ A + v for all real measures A, and 0/x ^ (2v for all real scalars a *» 0. (For a study of this order relation, see (13.15).) (13.3.2) Let VL be a (complex) measure on X. Then there exists a smallest positive measure p on X such that |//(/)| g p(\f\)for allfe Jf C(X). For every positive measure v such that |//(/)| ^ v(|/|) for all/e the relations g £ 0 and \h\ ^ g (g, h e JTC(X)) imply |ju(A)| g v(|A|) £ v(#). We shall show that there exists a positive measure p on X such that, for each function/^ 0 in JTR(X), we have (13.3.2.1) p(/)= sup This p will then clearly satisfy the conditions of (13.3.2). To begin with, we remark that the right-hand side of (13.3.2.1) is finite; for if K = Supp(/), then Supp(#) <= K, and whenever \g\ ^/, by virtue of (13.1.2). Also it is clear that, for any real scalar a g: 0, we have p(af) = ap(f). We shall show next that, if /t and /2 are two functions ^ 0 belonging to JTR(X), then (13-3.2.2) p(/i+/2) = p(/i) For each fi > 0, there exists g{ e «^TCW such \9i\<,ft and Multiplying g{ by a complex number with absolute value 1, we may assume that /<#,-) = \n(fft)\, and then we have and since |^ +^2I Sa/i +/2 we have p(/t +/2) ^ p(/j) + p(/2) - 2e. Since e is arbitrary, it follows that p(/i) + p(/2) ^ p(/i H-/2). On the other hand, let h e Jf C(X) be such that \h\ ^/i -f /2 . Let ht be the function which is equal to hfiKfi -f-/2) at the points x where /i(x) +/2(x) ^ 0 and is zero3.1.9) Let (Ua)a6l be an open covering ofX. For each a el let ua be a